Lecture 1 Notes: Intro to Course
Course overview
In this course, we will mainly be concerned with the following problems:
u = 0, i.e.
Harmonic functions
n
i
Dirichlet problem: ( R )
1)
= 0.
u
ii
,
u =0
u =
x ,
, x .
n+1
u, u : R R.
2) Heat equation: ut =
Lecture 2 Notes: Limit Theorem
Mean Value Theorem
2
n
u = 0 in , and B = B(y, R) , then
Theorem 1 Suppose R , uC (),
1
nnR
u(y) =
Proof:By Greens formula, forr(0, R),
1
n1
B uds =
u
Br
u
0=
Br
ds =
n1
Br udx = 0. Thus
(y + r)ds
Sn1 r
=r
r
(y + r)d
Sn1 u(y
Lecture 6
Notes: Kelvin
Transformati
ons
Last time: In spherical coordinates,u(x) =U(r, ),
n 1 Ur +
u(r, ) = Urr +
1
2
Sn1 U.
r
r
If U(r, ) = f(r)B(), then
u= (f + f )B() + f
rr r r
r2
n11
Sn1 B.
n1
Proposition 1 Eigenvalues of Sn1 are k(k+n2), where k0,
Lecture 3 Notes: General Domains
Definition of Greens function for general domains.
2
1
Suppose u C () C (), then for y , the Green Representation formula tells
us
(u (x y) (x y) )d +
u(y) =
(x y)udx.
u
density f.
y)f(x)dx is called Newtonian Potential w
Lecture 5
Notes:
Harmonic
Functions
Removable Singularity Theorem
Theorem 1 Let u be harmonic in \ cfw_x0, if
o( x x 2n)
| 0|
o(ln |x x0|)
u(x) =
,
n > 2,
,
n=2
as xx0, then u extends to a harmonic function in .
Proof: Without loss of generality, we can a
Lecture 4 Notes: Harmonic Variables
MVP + integrable harmonic
1
Theorem 1 Suppose uL loc, then u is harmonic u satisfies MVP on .
n
Proof: TakeC functionon R with properties: (a)Supp()B(0,1); (b)0;
(c) is radical, i.e. (x) = (|x|); and (d) B(0,1) (x)dx =
Lecture 8 Notes: Nonlinear Equations
Quasilinear equations (minimal surface equation)
n
n+1
For any f : R R, the graph of f is cfw_(x, f(x) R
.
th
on the i slot. So the
The tangents of the graph is (0,
, 1, 0, , 0, fi), where 1 is 1
normal vector is (f,
Lecture 9 Notes: Uniform Mathematics
A Priori Estimates for Poissons Equation.
Recall that N f =
Newtonian Potential of f .
(x y)f (y)dy is called the
1
(), and = N f is the Newto-
Proposition 1 Suppose is bounded domain, fL
1
n
nian potential of f . The
Lecture 7 Notes: Elliptic Operations
Weak maximum principle for linear elliptic operators
Now we consider the more general dierential operators
ij
i
L = a (x)Dij + b (x)Di + c(x),
2
i.e., for any C function u,
2
ij
i
u(x)
Lu = a (x)
+ b (x)
i j
x x
ij
u(
Lecture 10 Notes: Continuity
2
C estimates
Proposition 1 Let f be bounded and fC () (locally), (x) =
(x
2
be the Newtonian potential of f . Then C (), =f and
Dij (x y)(f (y) f (x)dy f (x)
Dij (x) =
y)f (y)dy
Di(x y)j dy ,
0
0
2
where 0 is nice domain (fo